Anders Schlichtkrull

Formalization of the Resolution Calculus for First-Order Logic

A formalization in Isabelle/HOL of the resolution calculus for first-order logic is presented. Its soundness and completeness are formally proven using the substitution lemma, semantic trees, Herbrand’s theorem, and the lifting lemma. In contrast to previous formalizations of resolution, it considers first-order logic with full first-order terms, instead of the propositional case.

Formalization of Bachmair and Ganzinger’s Ordered Resolution Prover

We present a formalization of the first half of Bachmair and Ganzinger’s chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies several of the fine points in the chapter’s text, emphasizing the value of formal proofs in the field of automated reasoning.

Natural Deduction and the Isabelle Proof Assistant

We describe our Natural Deduction Assistant (NaDeA) and the interfaces between the Isabelle proof assistant and NaDeA. In particular, we explain how NaDeA, using a generated prover that has been verified in Isabelle, provides feedback to the student, and also how NaDeA, for each formula proved by the student, provides a generated theorem that can be verified in Isabelle.

Formalizing Bachmair and Ganzinger's Ordered Resolution Prover

We present a formalization of the first half of Bachmair and Ganzinger’s chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies several of the fine points in the chapter’s text, emphasizing the value of formal proofs in the field of automated reasoning.

Programming and verifying a declarative first-order prover in Isabelle/HOL

We certify in the proof assistant Isabelle/HOL the soundness of a declarative first-order prover with equality. The LCF-style prover is a translation we have made, to Standard ML, of a prover in John Harrison’s Handbook of Practical Logic and Automated Reasoning. We certify it by replacing its kernel with a certified version that we program, certify and generate code from; all in Isabelle/HOL. In a declarative proof each step of the proof is declared, similar to the sentences in a thorough paper proof. The prover allows proofs to mix the declarative style with automatic theorem proving by using a tableau prover. Our motivation is teaching how automated and declarative provers work and how they are used. The prover allows studying concrete code and a formal verification of correctness. We show examples of proofs and how they are made in the prover. The entire development runs in Isabelle’s ML environment as an interactive application or can be used standalone in OCaml or Standard ML (or in other functional programming languages like Haskell and Scala with some additional work).

Formalizing a Paraconsistent Logic in the Isabelle Proof Assistant

We present a formalization of a so-called paraconsistent logic that avoids the catastrophic explosiveness of inconsistency in classical logic. The paraconsistent logic has a countably infinite number of non-classical truth values. We show how to use the proof assistant Isabelle to formally prove theorems in the logic as well as meta-theorems about the logic. In particular, we formalize a meta-theorem that allows us to reduce the infinite number of truth values to a finite number of truth values, for a given formula, and we use this result in a formalization of a small case study.